We are given the following unconstrained optimization problem:
$$ \mathbf{x}^* = \arg \min_{\mathbf{x}} f(\mathbf{x}) $$where \( f: \mathbb{R}^n \to \mathbb{R} \) is a differentiable objective function, and \( \mathbf{x} \in \mathbb{R}^n \) is the vector of decision variables.
The first-order necessary conditions for optimality are given by:
$$ \nabla f(\mathbf{x}^*) = 0 $$where \( \nabla f(\mathbf{x}) \) denotes the gradient of \( f(\mathbf{x}) \).